**DDA2001, April2001**

*Session 5. Satellites*

Tuesday, 8:30-10:20am
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## [5.03] Tidal Amplitude for a Self-gravitating, Compressible Sphere

*T.A. Hurford, R. Greenberg (Univ. of Arizona, LPL)*

Most modern evaluations of tidal amplitude derive from the
approach presented by Love [1]. Love's analysis for a
homogeneous sphere assumed an incompressible material, which
required introduction of a non-rigorously justified pressure
term. We solve the more general case of arbitrary
compressibility, which allows for a more straightforward
derivation. We find the h_{2} love number of a body of
radius R, density \rho, and surface gravity g to be
\begin{equation}\label{eq:hcomp} h_{2} =
\Bigg[\frac{5}{2}/1+\frac{19 \mu}{2 \rho g R}\Bigg]
\Bigg{ 2 \rho g R (35+28\frac{\mu}{\lambda}) + 19 \mu
(35+28\frac{\mu}{\lambda})/2 \rho g R
(35+31\frac{\mu}{\lambda}) + 19 \mu
(35+\frac{490}{19}\frac{\mu}{\lambda})\Bigg} \nonumber
\end{equation} \lambda the Lamé constant. This h_{2}
is the product of Love's expression for h_{2} (in square
brackets) and a ``compressibility-correction'' factor (in
{} brackets). Unlike Love's expression, this result is
valid for any degree of compressibility (i.e. any
\lambda). For the incompressible case (\lambda
arrow \infty) the correction factor approaches 1,
so that h_{2} matches the classical form given by Love.

In reality, of course, materials are not incompressible and
the difference between our solution and Love's is
significant. Assuming that the elastic terms dominate over
the gravitational contribution (i.e. 19 \mu/(2 \rho g R)
\gg 1), our solution can be ~7% percent larger than
Love's solution for large \mu/\lambda. If the gravity
dominates (i.e. 19 \mu/(2 \rho g R) \ll 1), our solution
is ~10 % smaller than Love's solution for large
\mu/\lambda. For example, a rocky body (\mu/\lambda ~
1 [2]), Earth-sized (19\mu/(2 \rho g R) ~1) body,
h_{2} would be reduced by about 1% from the classical
formula. Similarly, under some circumstances the l_{2}
Love number for a uniform sphere could be 22% smaller
than Love's evaluation. \\ [1] Love, A.E.H., \emph{A
Treatise on the Mathematical Theory of Elasticity}, New York
Dover Publications, 1944 \\ [2] Kaula, W.M., \emph{An
Introduction to Planetary Physics: The Terrestrial Planets},
John Wiley & Sons, Inc., 1968

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